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Topic: Few questions on forcing, large cardinals
Replies: 17   Last Post: Mar 30, 2013 1:21 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Few questions on forcing, large cardinals
Posted: Mar 23, 2013 8:01 PM
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On 3/23/2013 6:46 PM, Ross A. Finlayson wrote:
> On Mar 23, 3:34 pm, fom <fomJ...@nyms.net> wrote:
>> On 3/23/2013 5:09 PM, Ross A. Finlayson wrote:
>>
>>
>>
>>
>>
>>
>>
>>
>>

>>> On Mar 23, 2:44 pm, fom <fomJ...@nyms.net> wrote:
>>>> On 3/23/2013 4:34 PM, Ross A. Finlayson wrote:
>>
>>>>> In a sense, infinity _is_ the numbers. Start from even more
>>>>> fundamental objects than natural numbers as elements. Like the
>>>>> numbers, they are as different as they can be and as same as they can
>>>>> be, where they are each different in not being any other and each same
>>>>> in being defined by that difference. There's no stop to that, it's
>>>>> gone on, forever. Then, in a way like when you look into the void, it
>>>>> looks into you

>>
>>>> Is this your way of saying that if you look
>>>> into the void, you and the void become one?

>>
>>>> http://en.wikipedia.org/wiki/Cantor-Bernstein-Schroeder_theorem
>>
>>>> Just kiddding....
>>
>>>> I think Cantor would appreciate your sentiment that
>>>> the numbers of Cantor's paradise are more fundamental
>>>> than those of Kronecker's torment.

>>
>>> I wouldn't say that infinity, even in the numbers, is either of those
>>> things. In ZF, Infinity is _axiomatized_ to be an inductive set, and
>>> a well-founded/regular one, that's not a given. Calling that the
>>> universe, Russell's comment is that it would contain itself.

>>
>>> There's a case for induction, as it were, that each case exists. Then
>>> it is to be of deduction, not fiat by axiomatization, from simple
>>> principles of constancy and variety, the continuum.

>>
>>> In a theory with sets as primary objects, a set theory and a pure set
>>> theory, numbers would be very rich objects indeed, as not just
>>> individual elements by their elements, but all relations of numbers.
>>> Set theory (well-founded, as it were, regular or that objects are
>>> transitively closed) is at once over-simplification, to talk about
>>> anything besides sets, and over-complexification, to talk about itself
>>> when any universal statement is in the meta.

>>
>>> There are no numbers in a pure set theory. To call the natural
>>> integers a set, it contains only numbers, for the Platonists: elements
>>> of the structure, of numbers, as: none exist in a void.

>>
>> It is odd. In some sense, modern mathematics actually
>> treats its objects as urelements relative to set theory.
>> Looking at Hilbert, he makes statements whereby his formalism
>> is intended to supersede the class-based constructions of
>> Dedekind.
>>
>> Your frank statement that a set is not a number reflects
>> that sentiment.

>
> Particular finite sets are called ordinals, set-theoretic operations
> on them are defined that give the same results as Presburger/Peano
> arithmetic of the natural integers. The negative integers aren't
> simply the complement as in finite-word-width machine arithmetic, but
> again simple enough operations on sets (with the only ur-element being
> the empty set) give a "model" of the integers. Rationals are defined
> simply enough as equivalence classes over any pairs of integers,
> besides zeros, the reals then see the Least Upper Bound as axiom.
> These are all to match number-theoretic features, and largely suffice
> for integers and rational numbers, but not so obviously do sets
> suffice to represent thusly elements (and all of) the continuum of
> real numbers.
>
> Then, though, to call the empty set the number zero: wouldn't that be
> the number zero wherever there's an empty set? Building upwards to
> have particular sets for each of of the finite integers: then to
> build the numbers as sets, is to build all the relations of the
> numbers as sets, not just as to a set-theoretic model of only that set
> of numbers' operations: but of all instances, besides the schema.
> Where the ur-element is any thing, it so implies all other things,
> and is so implied. The collection and aggregates of sets or
> categorization or refinement of types or partition or bounding of
> division, are all of the same corpus.
>
> Here back to the questions as above:
>
> 1) is not forcing simply transfinite Dirichlet box?


I am not sure what you mean by this.

However, forcing might be better thought of as comparable
to Euclid's proof that there is no greatest prime.

> 2) are there any results due transfinite cardinals, not of transfinite
> cardinals?


The Borel hierarchy is defined in terms of the first
uncountable ordinal. Hence, results in descriptive
set theory that depend on that definition may count.

I do not have enough knowledge of that branch of
study to comment further.

> 3) is not an irregular model of ZF non-well-founded?

What is your definition of irregular?

> 4) does not a model of ZF contain itself?

There are relativizations of models. So, one question
in set theory is whether

HOD=HOD^HOD

where HOD are the hereditarily ordinal-defined
sets and HOD^HOD is HOD relativized within itself.

In this sense, models may have representations
within themselves. But, once again, expertise
is lacking here.

> 5) is ZF not a model of itself?

ZF is an axiomatization. The question is not
well-construed.






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